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  <div class="header">
    <h1 class="chapter_number">
      <a href="">CHAPTER 4</a>
    </h1>
    <h1 class="chapter_title"><a href="">A visual proof that neural nets can compute any function</a></h1>
  </div>
  <div class="section">
    <div id="toc">
      <p class="toc_title"><a href="index.html">Neural Networks and Deep Learning</a></p>
      <p class="toc_not_mainchapter"><a href="about.html">What this book is about</a></p>
      <p class="toc_not_mainchapter"><a href="exercises_and_problems.html">On the exercises and problems</a></p>
      <p class='toc_mainchapter'><a id="toc_using_neural_nets_to_recognize_handwritten_digits_reveal" class="toc_reveal"
          onMouseOver="this.style.borderBottom='1px solid #2A6EA6';" onMouseOut="this.style.borderBottom='0px';"><img
            id="toc_img_using_neural_nets_to_recognize_handwritten_digits" src="images/arrow.png" width="15px"></a><a
          href="chap1.html">Using neural nets to recognize handwritten digits</a>
      <div id="toc_using_neural_nets_to_recognize_handwritten_digits" style="display: none;">
        <p class="toc_section">
        <ul><a href="chap1.html#perceptrons">
            <li>Perceptrons</li>
          </a><a href="chap1.html#sigmoid_neurons">
            <li>Sigmoid neurons</li>
          </a><a href="chap1.html#the_architecture_of_neural_networks">
            <li>The architecture of neural networks</li>
          </a><a href="chap1.html#a_simple_network_to_classify_handwritten_digits">
            <li>A simple network to classify handwritten digits</li>
          </a><a href="chap1.html#learning_with_gradient_descent">
            <li>Learning with gradient descent</li>
          </a><a href="chap1.html#implementing_our_network_to_classify_digits">
            <li>Implementing our network to classify digits</li>
          </a><a href="chap1.html#toward_deep_learning">
            <li>Toward deep learning</li>
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      <p class='toc_mainchapter'><a id="toc_how_the_backpropagation_algorithm_works_reveal" class="toc_reveal"
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          href="chap2.html">How the backpropagation algorithm works</a>
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        <p class="toc_section">
        <ul><a href="chap2.html#warm_up_a_fast_matrix-based_approach_to_computing_the_output
_from_a_neural_network">
            <li>Warm up: a fast matrix-based approach to computing the output
              from a neural network</li>
          </a><a href="chap2.html#the_two_assumptions_we_need_about_the_cost_function">
            <li>The two assumptions we need about the cost function</li>
          </a><a href="chap2.html#the_hadamard_product_$s_\odot_t$">
            <li>The Hadamard product, $s \odot t$</li>
          </a><a href="chap2.html#the_four_fundamental_equations_behind_backpropagation">
            <li>The four fundamental equations behind backpropagation</li>
          </a><a href="chap2.html#proof_of_the_four_fundamental_equations_(optional)">
            <li>Proof of the four fundamental equations (optional)</li>
          </a><a href="chap2.html#the_backpropagation_algorithm">
            <li>The backpropagation algorithm</li>
          </a><a href="chap2.html#the_code_for_backpropagation">
            <li>The code for backpropagation</li>
          </a><a href="chap2.html#in_what_sense_is_backpropagation_a_fast_algorithm">
            <li>In what sense is backpropagation a fast algorithm?</li>
          </a><a href="chap2.html#backpropagation_the_big_picture">
            <li>Backpropagation: the big picture</li>
          </a></ul>
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      <p class='toc_mainchapter'><a id="toc_improving_the_way_neural_networks_learn_reveal" class="toc_reveal"
          onMouseOver="this.style.borderBottom='1px solid #2A6EA6';" onMouseOut="this.style.borderBottom='0px';"><img
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          href="chap3.html">Improving the way neural networks learn</a>
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        <p class="toc_section">
        <ul><a href="chap3.html#the_cross-entropy_cost_function">
            <li>The cross-entropy cost function</li>
          </a><a href="chap3.html#overfitting_and_regularization">
            <li>Overfitting and regularization</li>
          </a><a href="chap3.html#weight_initialization">
            <li>Weight initialization</li>
          </a><a href="chap3.html#handwriting_recognition_revisited_the_code">
            <li>Handwriting recognition revisited: the code</li>
          </a><a href="chap3.html#how_to_choose_a_neural_network's_hyper-parameters">
            <li>How to choose a neural network's hyper-parameters?</li>
          </a><a href="chap3.html#other_techniques">
            <li>Other techniques</li>
          </a></ul>
        </p>
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      <p class='toc_mainchapter'><a id="toc_a_visual_proof_that_neural_nets_can_compute_any_function_reveal"
          class="toc_reveal" onMouseOver="this.style.borderBottom='1px solid #2A6EA6';"
          onMouseOut="this.style.borderBottom='0px';"><img
            id="toc_img_a_visual_proof_that_neural_nets_can_compute_any_function" src="images/arrow.png"
            width="15px"></a><a href="chap4.html">A visual proof that neural nets can compute any function</a>
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        <p class="toc_section">
        <ul><a href="chap4.html#two_caveats">
            <li>Two caveats</li>
          </a><a href="chap4.html#universality_with_one_input_and_one_output">
            <li>Universality with one input and one output</li>
          </a><a href="chap4.html#many_input_variables">
            <li>Many input variables</li>
          </a><a href="chap4.html#extension_beyond_sigmoid_neurons">
            <li>Extension beyond sigmoid neurons</li>
          </a><a href="chap4.html#fixing_up_the_step_functions">
            <li>Fixing up the step functions</li>
          </a><a href="chap4.html#conclusion">
            <li>Conclusion</li>
          </a></ul>
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            id="toc_img_why_are_deep_neural_networks_hard_to_train" src="images/arrow.png" width="15px"></a><a
          href="chap5.html">Why are deep neural networks hard to train?</a>
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        <p class="toc_section">
        <ul><a href="chap5.html#the_vanishing_gradient_problem">
            <li>The vanishing gradient problem</li>
          </a><a href="chap5.html#what's_causing_the_vanishing_gradient_problem_unstable_gradients_in_deep_neural_nets">
            <li>What's causing the vanishing gradient problem? Unstable gradients in deep neural nets</li>
          </a><a href="chap5.html#unstable_gradients_in_more_complex_networks">
            <li>Unstable gradients in more complex networks</li>
          </a><a href="chap5.html#other_obstacles_to_deep_learning">
            <li>Other obstacles to deep learning</li>
          </a></ul>
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      <p class='toc_mainchapter'><a id="toc_deep_learning_reveal" class="toc_reveal"
          onMouseOver="this.style.borderBottom='1px solid #2A6EA6';" onMouseOut="this.style.borderBottom='0px';"><img
            id="toc_img_deep_learning" src="images/arrow.png" width="15px"></a><a href="chap6.html">Deep learning</a>
      <div id="toc_deep_learning" style="display: none;">
        <p class="toc_section">
        <ul><a href="chap6.html#introducing_convolutional_networks">
            <li>Introducing convolutional networks</li>
          </a><a href="chap6.html#convolutional_neural_networks_in_practice">
            <li>Convolutional neural networks in practice</li>
          </a><a href="chap6.html#the_code_for_our_convolutional_networks">
            <li>The code for our convolutional networks</li>
          </a><a href="chap6.html#recent_progress_in_image_recognition">
            <li>Recent progress in image recognition</li>
          </a><a href="chap6.html#other_approaches_to_deep_neural_nets">
            <li>Other approaches to deep neural nets</li>
          </a><a href="chap6.html#on_the_future_of_neural_networks">
            <li>On the future of neural networks</li>
          </a></ul>
        </p>
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      <p class="toc_not_mainchapter"><a href="sai.html">Appendix: Is there a <em>simple</em> algorithm for
          intelligence?</a></p>
      <p class="toc_not_mainchapter"><a href="acknowledgements.html">Acknowledgements</a></p>
      <p class="toc_not_mainchapter"><a href="faq.html">Frequently Asked Questions</a></p>
      <hr>
      <span class="sidebar_title">Resources</span>

      <p class="sidebar"><a href="https://twitter.com/michael_nielsen">Michael Nielsen on Twitter</a></p>

      <p class="sidebar"><a href="faq.html">Book FAQ</a></p>

      <p class="sidebar">
        <a href="https://github.com/mnielsen/neural-networks-and-deep-learning">Code repository</a>
      </p>

      <p class="sidebar">
        <a href="http://eepurl.com/0Xxjb">Michael Nielsen's project announcement mailing list</a>
      </p>

      <p class="sidebar"> <a href="http://www.deeplearningbook.org/">Deep Learning</a>, book by Ian
        Goodfellow, Yoshua Bengio, and Aaron Courville</p>

      <p class="sidebar"><a href="http://cognitivemedium.com">cognitivemedium.com</a></p>

      <hr>
      <a href="http://michaelnielsen.org"><img src="assets/Michael_Nielsen_Web_Small.jpg" width="160px"
          style="border-style: none;" /></a>

      <p class="sidebar">
        By <a href="http://michaelnielsen.org">Michael Nielsen</a> / Dec 2019
      </p>
    </div>
    </p>
    <p>One of the most striking facts about neural networks is that they can
      compute any function at all. That is, suppose someone hands you some
      complicated, wiggly function, $f(x)$:</p>
    <p>
      <center><canvas id="function" width="300" height="300"></canvas></center>
    </p>
    <p><a id="basic_network_precursor"></a> No matter what the
      function, there is guaranteed to be a neural network so that for every
      possible input, $x$, the value $f(x)$ (or some close approximation) is
      output from the network, e.g.:</p>
    <p>
      <center><canvas id="basic_network" width="350" height="220"></canvas></center>
    </p>
    <p>This result holds even if the function has many inputs, $f = f(x_1,
      \ldots, x_m)$, and many outputs. For instance, here's a network
      computing a function with $m = 3$ inputs and $n = 2$ outputs:</p>
    <p>
      <center><canvas id="vector_valued_network" width="450" height="370"></canvas></center>
    </p>
    <p>This result tells us that neural networks have a kind of
      <em>universality</em>. No matter what function we want to compute, we
      know that there is a neural network which can do the job.
    </p>
    <p>What's more, this universality theorem holds even if we restrict our
      networks to have just a single layer intermediate between the input
      and the output neurons - a so-called single hidden layer. So even
      very simple network architectures can be extremely powerful.</p>
    <p>The universality theorem is well known by people who use neural
      networks. But why it's true is not so widely understood. Most of the
      explanations available are quite technical. For instance, one of the
      original papers proving the
      result*<span class="marginnote">
        *<a href="http://www.dartmouth.edu/&#126;gvc/Cybenko_MCSS.pdf">Approximation
          by superpositions of a sigmoidal function</a>, by George Cybenko
        (1989). The result was very much in the air at the time, and
        several groups proved closely related results. Cybenko's paper
        contains a useful discussion of much of that work. Another
        important early paper is
        <a href="http://www.sciencedirect.com/science/article/pii/0893608089900208">Multilayer
          feedforward networks are universal approximators</a>, by Kurt Hornik,
        Maxwell Stinchcombe, and Halbert White (1989). This paper uses the
        Stone-Weierstrass theorem to arrive at similar results.</span> did so
      using the Hahn-Banach theorem, the Riesz Representation theorem, and
      some Fourier analysis. If you're a mathematician the argument is not
      difficult to follow, but it's not so easy for most people. That's a
      pity, since the underlying reasons for universality are simple and
      beautiful.</p>
    <p>In this chapter I give a simple and mostly visual explanation of the
      universality theorem. We'll go step by step through the underlying
      ideas. You'll understand why it's true that neural networks can
      compute any function. You'll understand some of the limitations of
      the result. And you'll understand how the result relates to deep
      neural networks.</p>
    <p>To follow the material in the chapter, you do not need to have read
      earlier chapters in this book. Instead, the chapter is structured to
      be enjoyable as a self-contained essay. Provided you have just a
      little basic familiarity with neural networks, you should be able to
      follow the explanation. I will, however, provide occasional links to
      earlier material, to help fill in any gaps in your knowledge.</p>
    <p></p>
    <p></p>
    <p></p>
    <p>Universality theorems are a commonplace in computer science, so much
      so that we sometimes forget how astonishing they are. But it's worth
      reminding ourselves: the ability to compute an arbitrary function is
      truly remarkable. Almost any process you can imagine can be thought
      of as function computation. Consider the problem of naming a piece of
      music based on a short sample of the piece. That can be thought of as
      computing a function. Or consider the problem of translating a
      Chinese text into English. Again, that can be thought of as computing
      a function*<span class="marginnote">
        *Actually, computing one of many functions, since
        there are often many acceptable translations of a given piece of
        text.</span>. Or consider the problem of taking an mp4 movie file and
      generating a description of the plot of the movie, and a discussion of
      the quality of the acting. Again, that can be thought of as a kind of
      function computation*<span class="marginnote">
        *Ditto the remark about translation and
        there being many possible functions.</span>. Universality means that, in
      principle, neural networks can do all these things and many more.</p>
    <p>Of course, just because we know a neural network exists that can (say)
      translate Chinese text into English, that doesn't mean we have good
      techniques for constructing or even recognizing such a network. This
      limitation applies also to traditional universality theorems for
      models such as Boolean circuits. But, as we've seen earlier in the
      book, neural networks have powerful algorithms for learning functions.
      That combination of learning algorithms + universality is an
      attractive mix. Up to now, the book has focused on the learning
      algorithms. In this chapter, we focus on universality, and what it
      means.</p>
    <p>
    <h3><a name="two_caveats"></a><a href="#two_caveats">Two caveats</a></h3>
    </p>
    <p>Before explaining why the universality theorem is true, I want to
      mention two caveats to the informal statement "a neural network can
      compute any function".</p>
    <p>First, this doesn't mean that a network can be used to <em>exactly</em>
      compute any function. Rather, we can get an <em>approximation</em>
      that is as good as we want. By increasing the number of hidden
      neurons we can improve the approximation. For instance,
      <a href="#basic_network_precursor">earlier</a> I illustrated a network
      computing some function $f(x)$ using three hidden neurons. For most
      functions only a low-quality approximation will be possible using
      three hidden neurons. By increasing the number of hidden neurons
      (say, to five) we can typically get a better approximation:
    </p>
    <p>
      <center><canvas id="bigger_network" width="350" height="380"></canvas></center>
    </p>
    <p>And we can do still better by further increasing the number of hidden
      neurons. </p>
    <p>To make this statement more precise, suppose we're given a function
      $f(x)$ which we'd like to compute to within some desired accuracy
      $\epsilon > 0$. The guarantee is that by using enough hidden neurons
      we can always find a neural network whose output $g(x)$ satisfies
      $|g(x) - f(x)| < \epsilon$, for all inputs $x$. In other words, the approximation will be good to within the
        desired accuracy for every possible input.</p>
        <p>The second caveat is that the class of functions which can be
          approximated in the way described are the <em>continuous</em> functions.
          If a function is discontinuous, i.e., makes sudden, sharp jumps, then
          it won't in general be possible to approximate using a neural net.
          This is not surprising, since our neural networks compute continuous
          functions of their input. However, even if the function we'd really
          like to compute is discontinuous, it's often the case that a
          continuous approximation is good enough. If that's so, then we can
          use a neural network. In practice, this is not usually an important
          limitation.</p>
        <p>Summing up, a more precise statement of the universality theorem is
          that neural networks with a single hidden layer can be used to
          approximate any continuous function to any desired precision. In this
          chapter we'll actually prove a slightly weaker version of this result,
          using two hidden layers instead of one. In the problems I'll briefly
          outline how the explanation can, with a few tweaks, be adapted to give
          a proof which uses only a single hidden layer.

        <h3><a name="universality_with_one_input_and_one_output"></a><a
            href="#universality_with_one_input_and_one_output">Universality with one input and one output</a></h3>
    </p>
    <p>To understand why the universality theorem is true, let's start by
      understanding how to construct a neural network which approximates a
      function with just one input and one output:</p>
    <p>
      <center><canvas id="function_2" width="300" height="300"></canvas></center>
    </p>
    <p>It turns out that this is the core of the problem of universality.
      Once we've understood this special case it's actually pretty easy to
      extend to functions with many inputs and many outputs.</p>
    <p>To build insight into how to construct a network to compute $f$, let's
      start with a network containing just a single hidden layer, with two
      hidden neurons, and an output layer containing a single output neuron:</p>
    <p>
      <center><canvas id="two_hidden_neurons" width="350" height="220"></canvas></center>
    </p>
    <p>To get a feel for how components in the network work, let's focus on
      the top hidden neuron. In the diagram below, click on the weight,
      $w$, and drag the mouse a little ways to the right to increase $w$.
      You can immediately see how the function computed by the top hidden
      neuron changes:</p>
    <p>
      <center><canvas id="basic_manipulation" width="600" height="285"></canvas></center>
    </p>
    <p>As we learnt <a href="chap1.html#sigmoid_neurons">earlier in the book</a>,
      what's being computed by the hidden neuron is $\sigma(wx + b)$, where
      $\sigma(z) \equiv 1/(1+e^{-z})$ is the sigmoid function. Up to now,
      we've made frequent use of this algebraic form. But for the proof of
      universality we will obtain more insight by ignoring the algebra
      entirely, and instead manipulating and observing the shape shown in
      the graph. This won't just give us a better feel for what's going on,
      it will also give us a proof*<span class="marginnote">
        *Strictly speaking, the visual
        approach I'm taking isn't what's traditionally thought of as a
        proof. But I believe the visual approach gives more insight into
        why the result is true than a traditional proof. And, of course,
        that kind of insight is the real purpose behind a proof.
        Occasionally, there will be small gaps in the reasoning I present:
        places where I make a visual argument that is plausible, but not
        quite rigorous. If this bothers you, then consider it a challenge
        to fill in the missing steps. But don't lose sight of the real
        purpose: to understand why the universality theorem is true.</span> of
      universality that applies to activation functions other than the
      sigmoid function.</p>
    <p>To get started on this proof, try clicking on the bias, $b$, in the
      diagram above, and dragging to the right to increase it. You'll see
      that as the bias increases the graph moves to the left, but its shape
      doesn't change.</p>
    <p>Next, click and drag to the left in order to decrease the bias.
      You'll see that as the bias decreases the graph moves to the right,
      but, again, its shape doesn't change.</p>
    <p>Next, decrease the weight to around $2$ or $3$. You'll see that as
      you decrease the weight, the curve broadens out. You might need to
      change the bias as well, in order to keep the curve in-frame.</p>
    <p>Finally, increase the weight up past $w = 100$. As you do, the curve
      gets steeper, until eventually it begins to look like a step function.
      Try to adjust the bias so the step occurs near $x = 0.3$. The
      following short clip shows what your result should look like. Click
      on the play button to play (or replay) the video:</p>
    <p>
      <!-- Based on http://worrydream.com/ScrubbingCalculator/, with minor changes -->
      <script type="text/javascript">
        function playVideo(name) {
          var div = $("#" + name)[0];
          div.style.backgroundColor = "transparent";
          div.style.cursor = "default";
          div.getElementsByTagName("img")[0].style.display = "none";
          var video = $("#v" + name)[0];
          video.play();
        }
        function videoEnded(name) {
          var div = document.getElementById(name);
          div.getElementsByTagName("img")[0].style.display = "block";
          div.style.backgroundColor = "white";
          div.style.opacity = 0.6;
          div.style.cursor = "pointer";
        }
      </script>
    <div>
      <div id="a" class="videoOverlay" style="width: 560px; height: 280px; opacity: 0.8" onclick="playVideo('a');">
        <img style="left: 210px; top: 75px;" src="images/play.png" width="128px">
      </div>
      <video id="va" width="560" height="280" preload onended="videoEnded('a');">
        <source type="video/webm" src="movies/create_step_function.webm">
        <source type="video/mp4" src="movies/create_step_function.mp4">
      </video>
    </div>
    </p>
    <p>We can simplify our analysis quite a bit by increasing the weight so
      much that the output really is a step function, to a very good
      approximation. Below I've plotted the output from the top hidden
      neuron when the weight is $w = 999$. Note that this plot is static,
      and you can't change parameters such as the weight.</p>
    <p><img src="images/high_weight_function.jpg"></p>
    <p>It's actually quite a bit easier to work with step functions than
      general sigmoid functions. The reason is that in the output layer we
      add up contributions from all the hidden neurons. It's easy to
      analyze the sum of a bunch of step functions, but rather more
      difficult to reason about what happens when you add up a bunch of
      sigmoid shaped curves. And so it makes things much easier to assume
      that our hidden neurons are outputting step functions. More
      concretely, we do this by fixing the weight $w$ to be some very large
      value, and then setting the position of the step by modifying the
      bias. Of course, treating the output as a step function is an
      approximation, but it's a very good approximation, and for now we'll
      treat it as exact. I'll come back later to discuss the impact of
      deviations from this approximation.</p>
    <p>At what value of $x$ does the step occur? Put another way, how does
      the position of the step depend upon the weight and bias?</p>
    <p>To answer this question, try modifying the weight and bias in the
      diagram above (you may need to scroll back a bit). Can you figure out
      how the position of the step depends on $w$ and $b$? With a little
      work you should be able to convince yourself that the position of the
      step is <em>proportional</em> to $b$, and <em>inversely proportional</em>
      to $w$.</p>
    <p>In fact, the step is at position $s = -b/w$, as you can see by
      modifying the weight and bias in the following diagram:</p>
    <p><canvas id="step" width="600" height="285"></canvas></p>
    <p>It will greatly simplify our lives to describe hidden neurons using
      just a single parameter, $s$, which is the step position, $s = -b/w$.
      Try modifying $s$ in the following diagram, in order to get used to
      the new parameterization:</p>
    <p><canvas id="step_parameterization" width="600" height="285"></canvas></p>
    <p>As noted above, we've implicitly set the weight $w$ on the input to be
      some large value - big enough that the step function is a very good
      approximation. We can easily convert a neuron parameterized in this
      way back into the conventional model, by choosing the bias $b = -w s$.</p>
    <p>Up to now we've been focusing on the output from just the top hidden
      neuron. Let's take a look at the behavior of the entire network. In
      particular, we'll suppose the hidden neurons are computing step
      functions parameterized by step points $s_1$ (top neuron) and $s_2$
      (bottom neuron). And they'll have respective output weights $w_1$ and
      $w_2$. Here's the network:</p>
    <p><canvas id="two_hn_network" width="600" height="285"></canvas></p>
    <p>What's being plotted on the right is the <em>weighted output</em> $w_1
      a_1 + w_2 a_2$ from the hidden layer. Here, $a_1$ and $a_2$ are the
      outputs from the top and bottom hidden neurons,
      respectively*<span class="marginnote">
        *Note, by the way, that the output from the whole
        network is $\sigma(w_1 a_1+w_2 a_2 + b)$, where $b$ is the bias on
        the output neuron. Obviously, this isn't the same as the weighted
        output from the hidden layer, which is what we're plotting here.
        We're going to focus on the weighted output from the hidden layer
        right now, and only later will we think about how that relates to
        the output from the whole network.</span>. These outputs are denoted with
      $a$s because they're often known as the neurons' <em>activations</em>.</p>
    <p>Try increasing and decreasing the step point $s_1$ of the top hidden
      neuron. Get a feel for how this changes the weighted output from the
      hidden layer.

      It's particularly worth understanding what happens when $s_1$ goes
      past $s_2$. You'll see that the graph changes shape when this
      happens, since we have moved from a situation where the top hidden
      neuron is the first to be activated to a situation where the bottom
      hidden neuron is the first to be activated.</p>
    <p>Similarly, try manipulating the step point $s_2$ of the bottom hidden
      neuron, and get a feel for how this changes the combined output from
      the hidden neurons.</p>
    <p>Try increasing and decreasing each of the output weights. Notice how
      this rescales the contribution from the respective hidden neurons.
      What happens when one of the weights is zero?</p>
    <p>Finally, try setting $w_1$ to be $0.8$ and $w_2$ to be $-0.8$. You
      get a "bump" function, which starts at point $s_1$, ends at point
      $s_2$, and has height $0.8$. For instance, the weighted output might
      look like this:</p>
    <p><img src="images/bump_function.jpg"></p>
    <p>Of course, we can rescale the bump to have any height at all. Let's
      use a single parameter, $h$, to denote the height. To reduce clutter
      I'll also remove the "$s_1 = \ldots$" and "$w_1 = \ldots$" notations.</p>
    <p><canvas id="bump_fn" width="600" height="285"></canvas></p>
    <p>Try changing the value of $h$ up and down, to see how the height of
      the bump changes. Try changing the height so it's negative, and
      observe what happens. And try changing the step points to see how
      that changes the shape of the bump.</p>
    <p>You'll notice, by the way, that we're using our neurons in a way that
      can be thought of not just in graphical terms, but in more
      conventional programming terms, as a kind of <tt>if-then-else</tt>
      statement, e.g.:</p>
    <p>
    <div class="highlight">
      <pre><span></span>    <span class="k">if</span> input &gt;<span class="o">=</span> step point:
        add <span class="m">1</span> to the weighted output
    <span class="k">else</span>:
        add <span class="m">0</span> to the weighted output
</pre>
    </div>
    </p>
    <p>For the most part I'm going to stick with the graphical point of view.
      But in what follows you may sometimes find it helpful to switch points
      of view, and think about things in terms of <tt>if-then-else</tt>.</p>
    <p>We can use our bump-making trick to get two bumps, by gluing two pairs
      of hidden neurons together into the same network:</p>
    <p><canvas id="double_bump" width="600" height="280"></canvas></p>
    <p>I've suppressed the weights here, simply writing the $h$ values for
      each pair of hidden neurons. Try increasing and decreasing both $h$
      values, and observe how it changes the graph. Move the bumps around
      by changing the step points.</p>
    <p>More generally, we can use this idea to get as many peaks as we want,
      of any height. In particular, we can divide the interval $[0, 1]$ up
      into a large number, $N$, of subintervals, and use $N$ pairs of hidden
      neurons to set up peaks of any desired height. Let's see how this
      works for $N = 5$. That's quite a few neurons, so I'm going to pack
      things in a bit. Apologies for the complexity of the diagram: I could
      hide the complexity by abstracting away further, but I think it's
      worth putting up with a little complexity, for the sake of getting a
      more concrete feel for how these networks work.</p>
    <p><canvas id="five_bumps" width="600" height="620"></canvas></p>
    <p>You can see that there are five pairs of hidden neurons. The step
      points for the respective pairs of neurons are $0, 1/5$, then $1/5,
      2/5$, and so on, out to $4/5, 5/5$. These values are fixed - they
      make it so we get five evenly spaced bumps on the graph.</p>
    <p>Each pair of neurons has a value of $h$ associated to it. Remember,
      the connections output from the neurons have weights $h$ and $-h$ (not
      marked). Click on one of the $h$ values, and drag the mouse to the
      right or left to change the value. As you do so, watch the function
      change. By changing the output weights we're actually
      <em>designing</em> the function!
    </p>
    <p>Contrariwise, try clicking on the graph, and dragging up or down to
      change the height of any of the bump functions. As you change the
      heights, you can see the corresponding change in $h$ values. And,
      although it's not shown, there is also a change in the corresponding
      output weights, which are $+h$ and $-h$.</p>
    <p>In other words, we can directly manipulate the function appearing in
      the graph on the right, and see that reflected in the $h$ values on
      the left. A fun thing to do is to hold the mouse button down and drag
      the mouse from one side of the graph to the other. As you do this you
      draw out a function, and get to watch the parameters in the neural
      network adapt.</p>
    <p>Time for a challenge.</p>
    <p>Let's think back to the function I plotted at the beginning of the
      chapter:</p>
    <p>
      <center> <canvas id="function_3" width="300" height="300"></canvas>
      </center>
    </p>
    <p>I didn't say it at the time, but what I plotted is actually the
      function
      <a class="displaced_anchor" name="eqtn113"></a>\begin{eqnarray}
      f(x) = 0.2+0.4 x^2+0.3x \sin(15 x) + 0.05 \cos(50 x),
      \tag{113}\end{eqnarray}
      plotted over $x$ from $0$ to $1$, and with the $y$ axis taking
      values from $0$ to $1$.
    </p>
    <p>That's obviously not a trivial function.</p>
    <p>You're going to figure out how to compute it using a neural network.</p>
    <p>In our networks above we've been analyzing the weighted combination
      $\sum_j w_j a_j$ output from the hidden neurons. We now know how to
      get a lot of control over this quantity. But, as I noted earlier,
      this quantity is not what's output from the network. What's output
      from the network is $\sigma(\sum_j w_j a_j + b)$ where $b$ is the bias
      on the output neuron. Is there some way we can achieve control over
      the actual output from the network?</p>
    <p>The solution is to design a neural network whose hidden layer has a
      weighted output given by $\sigma^{-1} \circ f(x)$, where $\sigma^{-1}$
      is just the inverse of the $\sigma$ function. That is, we want the
      weighted output from the hidden layer to be:</p>
    <p>
      <center> <canvas id="inverted_function" width="340" height="300"></canvas> </center>
    </p>
    <p>If we can do this, then the output from the network as a whole will be
      a good approximation to $f(x)$*<span class="marginnote">
        *Note that I have set the bias
        on the output neuron to $0$.</span>.</p>
    <p>Your challenge, then, is to design a neural network to approximate the
      goal function shown just above. To learn as much as possible, I want
      you to solve the problem twice. The first time, please click on the
      graph, directly adjusting the heights of the different bump functions.
      You should find it fairly easy to get a good match to the goal
      function. How well you're doing is measured by the <em>average
        deviation</em> between the goal function and the function the network is
      actually computing. Your challenge is to drive the average deviation
      as <em>low</em> as possible. You complete the challenge when you drive
      the average deviation to $0.40$ or below.</p>
    <p>Once you've done that, click on "Reset" to randomly re-initialize
      the bumps. The second time you solve the problem, resist the urge to
      click on the graph. Instead, modify the $h$ values on the left-hand
      side, and again attempt to drive the average deviation to $0.40$ or
      below.</p>
    <p><canvas id="design_function" width="600" height="620"></canvas></p>
    <p>You've now figured out all the elements necessary for the network to
      approximately compute the function $f(x)$! It's only a coarse
      approximation, but we could easily do much better, merely by
      increasing the number of pairs of hidden neurons, allowing more bumps.</p>
    <p>In particular, it's easy to convert all the data we have found back
      into the standard parameterization used for neural networks. Let me
      just recap quickly how that works.</p>
    <p>The first layer of weights all have some large, constant value, say $w
      = 1000$.</p>
    <p>The biases on the hidden neurons are just $b = -w s$. So, for
      instance, for the second hidden neuron $s = 0.2$ becomes $b = -1000
      \times 0.2 = -200$.</p>
    <p>The final layer of weights are determined by the $h$ values. So, for
      instance, the value you've chosen above for the first $h$, $h = $
      <span id="h" style="font-family: MJX_Main;"></span>, means that
      the output weights from the top two hidden neurons are <span id="w1" style="font-family: MJX_Main;"></span> and
      <span id="w2" style="font-family: MJX_Main;"></span>, respectively. And
      so on, for the entire layer of output weights.
    </p>
    <p>Finally, the bias on the output neuron is $0$.</p>
    <p>That's everything: we now have a complete description of a neural
      network which does a pretty good job computing our original goal
      function. And we understand how to improve the quality of the
      approximation by improving the number of hidden neurons.</p>
    <p>What's more, there was nothing special about our original goal
      function, $f(x) = 0.2+0.4 x^2+0.3 \sin(15 x) + 0.05 \cos(50 x)$. We
      could have used this procedure for any continuous function from $[0,
      1]$ to $[0, 1]$. In essence, we're using our single-layer neural
      networks to build a lookup table for the function. And we'll be able
      to build on this idea to provide a general proof of universality.</p>
    <p>
    <h3><a name="many_input_variables"></a><a href="#many_input_variables">Many input variables</a></h3>
    </p>
    <p>Let's extend our results to the case of many input variables. This
      sounds complicated, but all the ideas we need can be understood in the
      case of just two inputs. So let's address the two-input case.</p>
    <p>We'll start by considering what happens when we have two inputs to a
      neuron:</p>
    <p>
      <center> <canvas id="two_inputs" width="350" height="220"></canvas>
      </center>
    </p>
    <p>Here, we have inputs $x$ and $y$, with corresponding weights $w_1$ and
      $w_2$, and a bias $b$ on the neuron. Let's set the weight $w_2$ to
      $0$, and then play around with the first weight, $w_1$, and the bias,
      $b$, to see how they affect the output from the neuron:</p>
    <p>
      <script src="js/three.min.js"></script>
    </p>
    <p><canvas id="ti_graph" width="200" height="220"></canvas>
      <span id="ti_graph_3d" style="position: absolute; left: 260px;"></span>
    </p>
    <p>As you can see, with $w_2 = 0$ the input $y$ makes no difference to
      the output from the neuron. It's as though $x$ is the only input.</p>
    <p>Given this, what do you think happens when we increase the weight
      $w_1$ to $w_1 = 100$, with $w_2$ remaining $0$? If you don't
      immediately see the answer, ponder the question for a bit, and see if
      you can figure out what happens. Then try it out and see if you're
      right. I've shown what happens in the following movie:</p>
    <p>
    <div>
      <div id="b" class="videoOverlay" style="width: 460px; height: 252px; opacity: 0.8" onclick="playVideo('b');">
        <img style="left: 160px; top: 70px;" src="images/play.png" width="128px">
      </div>
      <video id="vb" width="460" height="252" preload onended="videoEnded('b');">
        <source type="video/mp4" src="movies/step_3d.mp4">
        <source type="video/webm" src="movies/step_3d.webm">
        </p>
        <p>
      </video>
    </div>
    </p>
    <p>Just as in our earlier discussion, as the input weight gets larger the
      output approaches a step function. The difference is that now the
      step function is in three dimensions. Also as before, we can move the
      location of the step point around by modifying the bias. The actual
      location of the step point is $s_x \equiv -b / w_1$.</p>
    <p>Let's redo the above using the position of the step as the parameter:</p>
    <p><canvas id="ti_graph_redux" width="200" height="220"></canvas> <span id="ti_graph_redux_3d" style="position: absolute; left:
260px;"></span></p>
    <p>Here, we assume the weight on the $x$ input has some large value
      - I've used $w_1 = 1000$ - and the weight $w_2 = 0$. The
      number on the neuron is the step point, and the little $x$ above the
      number reminds us that the step is in the $x$ direction.

      Of course, it's also possible to get a step function in the $y$
      direction, by making the weight on the $y$ input very large (say, $w_2
      = 1000$), and the weight on the $x$ equal to $0$, i.e., $w_1 = 0$:</p>
    <p><canvas id="y_step" width="200" height="220"></canvas> <span id="y_step_3d"
        style="position: absolute; left: 260px;"></span></p>
    <p>The number on the neuron is again the step point, and in this case the
      little $y$ above the number reminds us that the step is in the $y$
      direction. I could have explicitly marked the weights on the $x$ and
      $y$ inputs, but decided not to, since it would make the diagram rather
      cluttered. But do keep in mind that the little $y$ marker implicitly
      tells us that the $y$ weight is large, and the $x$ weight is $0$.</p>
    <p>We can use the step functions we've just constructed to compute a
      three-dimensional bump function. To do this, we use two neurons, each
      computing a step function in the $x$ direction. Then we combine those
      step functions with weight $h$ and $-h$, respectively, where $h$ is
      the desired height of the bump. It's all illustrated in the following
      diagram:</p>
    <p><canvas id="bump_3d" width="300" height="220"></canvas> <span id="bump_3d_graph"
        style="position: absolute; left: 360px;"></span></p>
    <p>Try changing the value of the height, $h$. Observe how it relates to
      the weights in the network. And see how it changes the height of the
      bump function on the right.</p>
    <p>Also, try changing the step point $0.30$ associated to the top hidden
      neuron. Witness how it changes the shape of the bump. What happens
      when you move it past the step point $0.70$ associated to the bottom
      hidden neuron?</p>
    <p>We've figured out how to make a bump function in the $x$ direction.
      Of course, we can easily make a bump function in the $y$ direction, by
      using two step functions in the $y$ direction. Recall that we do this
      by making the weight large on the $y$ input, and the weight $0$ on the
      $x$ input. Here's the result:</p>
    <p><canvas id="bump_3d_y" width="300" height="220"></canvas> <span id="bump_3d_y_graph"
        style="position: absolute; left: 360px;"></span></p>
    <p>This looks nearly identical to the earlier network! The only thing
      explicitly shown as changing is that there's now little $y$ markers on
      our hidden neurons. That reminds us that they're producing $y$ step
      functions, not $x$ step functions, and so the weight is very large on
      the $y$ input, and zero on the $x$ input, not vice versa. As before,
      I decided not to show this explicitly, in order to avoid clutter.</p>
    <p>Let's consider what happens when we add up two bump functions, one in
      the $x$ direction, the other in the $y$ direction, both of height $h$:</p>
    <p><canvas id="xy_bump" width="300" height="270"></canvas> <span id="xy_bump_3d"
        style="position: absolute; left: 360px;"></span></p>
    <p>To simplify the diagram I've dropped the connections with zero weight.
      For now, I've left in the little $x$ and $y$ markers on the hidden
      neurons, to remind you in what directions the bump functions are being
      computed. We'll drop even those markers later, since they're implied
      by the input variable.</p>
    <p>Try varying the parameter $h$. As you can see, this causes the output
      weights to change, and also the heights of both the $x$ and $y$ bump
      functions.</p>
    <p>What we've built looks a little like a <em>tower</em> function:</p>
    <p>
      <center> <span id="tower" style="position: absolute;"></span>
        <div style="height: 230px"></div>
      </center>
    </p>
    <p>If we could build such tower functions, then we could use them to
      approximate arbitrary functions, just by adding up many towers of
      different heights, and in different locations:</p>
    <p>
      <center> <span id="many_towers" style="position: absolute;"></span>
        <div style="height: 230px"></div>
      </center>
    </p>
    <p>Of course, we haven't yet figured out how to build a tower function.
      What we have constructed looks like a central tower, of height $2h$,
      with a surrounding plateau, of height $h$.</p>
    <p>But we can make a tower function. Remember that earlier we saw
      neurons can be used to implement a type of <tt>if-then-else</tt>
      statement:</p>
    <p>
    <div class="highlight">
      <pre><span></span>    <span class="k">if</span> input &gt;<span class="o">=</span> threshold: 
        output 1
    <span class="k">else</span>:
        output 0
</pre>
    </div>
    </p>
    <p>That was for a neuron with just a single input. What we want is to
      apply a similar idea to the combined output from the hidden neurons:</p>
    <p>
    <div class="highlight">
      <pre><span></span>    <span class="k">if</span> combined output from hidden neurons &gt;<span class="o">=</span> threshold:
        output 1
    <span class="k">else</span>:
        output 0
</pre>
    </div>
    </p>
    <p>If we choose the <tt>threshold</tt> appropriately - say, a value of
      $3h/2$, which is sandwiched between the height of the plateau and the
      height of the central tower - we could squash the plateau down to
      zero, and leave just the tower standing.</p>
    <p>Can you see how to do this? Try experimenting with the following
      network to figure it out. Note that we're now plotting the output
      from the entire network, not just the weighted output from the hidden
      layer. This means we add a bias term to the weighted output from the
      hidden layer, and apply the sigma function. Can you find values for
      $h$ and $b$ which produce a tower? This is a bit tricky, so if you
      think about this for a while and remain stuck, here's two hints: (1)
      To get the output neuron to show the right kind of <tt>if-then-else</tt>
      behaviour, we need the input weights (all $h$ or $-h$) to be large;
      and (2) the value of $b$ determines the scale of the
      <tt>if-then-else</tt> threshold.
    </p>
    <p><canvas id="tower_construction" width="300" height="270"></canvas>
      <span id="tower_construction_3d" style="position: absolute; left:
350px;"></span>
    </p>
    <p>With our initial parameters, the output looks like a flattened version
      of the earlier diagram, with its tower and plateau. To get the
      desired behaviour, we increase the parameter $h$ until it becomes
      large. That gives the <tt>if-then-else</tt> thresholding
      behaviour. Second, to get the threshold right, we'll choose $b
      \approx -3h/2$. Try it, and see how it works!</p>
    <p>Here's what it looks like, when we use $h = 10$:</p>
    <p>
    <div>
      <div id="c" class="videoOverlay" style="width: 556px; height: 284px; opacity: 0.8" onclick="playVideo('c');">
        <img style="left: 210px; top: 80px;" src="images/play.png" width="128px">
      </div>
      <video id="vc" width="556" height="284" preload onended="videoEnded('c');">
        <source type="video/mp4" src="movies/tower_construction.mp4">
        <source type="video/webm" src="movies/tower_construction.webm">
      </video>
    </div>
    </p>
    <p>Even for this relatively modest value of $h$, we get a pretty good
      tower function. And, of course, we can make it as good as we want by
      increasing $h$ still further, and keeping the bias as $b = -3h/2$.</p>
    <p>Let's try gluing two such networks together, in order to compute two
      different tower functions. To make the respective roles of the two
      sub-networks clear I've put them in separate boxes, below: each box
      computes a tower function, using the technique described above. The
      graph on the right shows the weighted output from the <em>second</em>
      hidden layer, that is, it's a weighted combination of tower functions.</p>
    <p><canvas id="the_two_towers" width="320" height="580"></canvas> <span id="the_two_towers_3d" style="position: absolute; left: 370px;
margin-top: 180px;"></span></p>
    <p>In particular, you can see that by modifying the weights in the final
      layer you can change the height of the output towers.</p>
    <p>The same idea can be used to compute as many towers as we like. We
      can also make them as thin as we like, and whatever height we like.
      As a result, we can ensure that the weighted output from the second
      hidden layer approximates any desired function of two variables:</p>
    <p>
      <center> <span id="many_towers_2" style="position: absolute;"></span>
        <div style="height: 230px"></div>
      </center>
    </p>
    <p>In particular, by making the weighted output from the second hidden
      layer a good approximation to $\sigma^{-1} \circ f$, we ensure the
      output from our network will be a good approximation to any desired
      function, $f$.</p>
    <p>What about functions of more than two variables?</p>
    <p>Let's try three variables $x_1, x_2, x_3$. The following network can
      be used to compute a tower function in four dimensions:</p>
    <p><canvas id="tower_n_dim" width="300" height="410"></canvas></p>
    <p>Here, the $x_1, x_2, x_3$ denote inputs to the network. The $s_1,
      t_1$ and so on are step points for neurons - that is, all the
      weights in the first layer are large, and the biases are set to give
      the step points $s_1, t_1, s_2, \ldots$. The weights in the second
      layer alternate $+h, -h$, where $h$ is some very large number. And
      the output bias is $-5h/2$.</p>
    <p>This network computes a function which is $1$ provided three
      conditions are met: $x_1$ is between $s_1$ and $t_1$; $x_2$ is between
      $s_2$ and $t_2$; and $x_3$ is between $s_3$ and $t_3$. The network is
      $0$ everywhere else. That is, it's a kind of tower which is $1$ in a
      little region of input space, and $0$ everywhere else.</p>
    <p>By gluing together many such networks we can get as many towers as we
      want, and so approximate an arbitrary function of three variables.
      Exactly the same idea works in $m$ dimensions. The only change needed
      is to make the output bias $(-m+1/2)h$, in order to get the right kind
      of sandwiching behavior to level the plateau.</p>
    <p>Okay, so we now know how to use neural networks to approximate a
      real-valued function of many variables. What about vector-valued
      functions $f(x_1, \ldots, x_m) \in R^n$? Of course, such a function
      can be regarded as just $n$ separate real-valued functions, $f^1(x_1,
      \ldots, x_m), f^2(x_1, \ldots, x_m)$, and so on. So we create a
      network approximating $f^1$, another network for $f^2$, and so on.
      And then we simply glue all the networks together. So that's also
      easy to cope with.</p>
    <p>
    <h4><a name="problem_863961"></a><a href="#problem_863961">Problem</a></h4>
    <ul>
      </p>
      <p>
        <li> We've seen how to use networks with two hidden layers to
          approximate an arbitrary function. Can you find a proof showing
          that it's possible with just a single hidden layer? As a hint, try
          working in the case of just two input variables, and showing that:
          (a) it's possible to get step functions not just in the $x$ or $y$
          directions, but in an arbitrary direction; (b) by adding up many of
          the constructions from part (a) it's possible to approximate a tower
          function which is circular in shape, rather than rectangular; (c)
          using these circular towers, it's possible to approximate an
          arbitrary function. To do part (c) it may help to use ideas from a
          <a href="#fixing_up_the_step_functions">bit later in this
            chapter</a>.
      </p>
      <p>
    </ul>
    </p>
    <p>
    <h3><a name="extension_beyond_sigmoid_neurons"></a><a href="#extension_beyond_sigmoid_neurons">Extension beyond
        sigmoid neurons</a></h3>
    </p>
    <p>We've proved that networks made up of sigmoid neurons can compute any
      function. Recall that in a sigmoid neuron the inputs $x_1, x_2,
      \ldots$ result in the output $\sigma(\sum_j w_j x_j + b)$, where $w_j$
      are the weights, $b$ is the bias, and $\sigma$ is the sigmoid
      function:</p>
    <p>
      <center> <canvas id="sigmoid" width="500" height="200"></canvas>
      </center>
    </p>
    <p>What if we consider a different type of neuron, one using some other
      activation function, $s(z)$:</p>
    <p>
      <center> <canvas id="sigmoid_like" width="500" height="200"></canvas>
      </center>
    </p>
    <p>That is, we'll assume that if our neurons has inputs $x_1, x_2,
      \ldots$, weights $w_1, w_2, \ldots$ and bias $b$, then the output is
      $s(\sum_j w_j x_j + b)$.</p>
    <p>We can use this activation function to get a step function, just as we
      did with the sigmoid. Try ramping up the weight in the following, say
      to $w = 100$:</p>
    <p><canvas id="ramping" width="600" height="200"></canvas></p>
    <p>Just as with the sigmoid, this causes the activation function to
      contract, and ultimately it becomes a very good approximation to a
      step function. Try changing the bias, and you'll see that we can set
      the position of the step to be wherever we choose. And so we can use
      all the same tricks as before to compute any desired function.</p>
    <p>What properties does $s(z)$ need to satisfy in order for this to work?
      We do need to assume that $s(z)$ is well-defined as $z \rightarrow
      -\infty$ and $z \rightarrow \infty$. These two limits are the two
      values taken on by our step function. We also need to assume that
      these limits are different from one another. If they weren't, there'd
      be no step, simply a flat graph! But provided the activation function
      $s(z)$ satisfies these properties, neurons based on such an activation
      function are universal for computation.</p>
    <p>
    <h4><a name="problems_963556"></a><a href="#problems_963556">Problems</a></h4>
    <ul>
      <li> Earlier in the book we met another type of neuron known as a <a
          href="chap3.html#other_models_of_artificial_neuron">rectified linear
          unit</a>. Explain why such neurons don't satisfy the conditions
        just given for universality. Find a proof of universality showing
        that rectified linear units are universal for computation.</p>
        <p>
      <li> Suppose we consider linear neurons, i.e., neurons with the
        activation function $s(z) = z$. Explain why linear neurons don't
        satisfy the conditions just given for universality. Show that such
        neurons can't be used to do universal computation.
    </ul>
    </p>
    <p>
    <h3><a name="fixing_up_the_step_functions"></a><a href="#fixing_up_the_step_functions">Fixing up the step
        functions</a></h3>
    </p>
    <p>Up to now, we've been assuming that our neurons can produce step
      functions exactly. That's a pretty good approximation, but it is only
      an approximation. In fact, there will be a narrow window of failure,
      illustrated in the following graph, in which the function behaves very
      differently from a step function:</p>
    <p><canvas id="failure" width="220" height="200"></canvas></p>
    <p>In these windows of failure the explanation I've given for
      universality will fail.</p>
    <p>Now, it's not a terrible failure. By making the weights input to the
      neurons big enough we can make these windows of failure as small as we
      like. Certainly, we can make the window much narrower than I've shown
      above - narrower, indeed, than our eye could see. So perhaps we
      might not worry too much about this problem.</p>
    <p>Nonetheless, it'd be nice to have some way of addressing the problem.</p>
    <p>In fact, the problem turns out to be easy to fix. Let's look at the
      fix for neural networks computing functions with just one input and
      one output. The same ideas work also to address the problem when
      there are more inputs and outputs.</p>
    <p>In particular, suppose we want our network to compute some function,
      $f$. As before, we do this by trying to design our network so that
      the weighted output from our hidden layer of neurons is $\sigma^{-1}
      \circ f(x)$:</p>
    <p>
      <center> <canvas id="inverted_function_2" width="340" height="300"></canvas> </center>
    </p>
    <p>If we were to do this using the technique described earlier, we'd use
      the hidden neurons to produce a sequence of bump functions:</p>
    <p>
      <center> <canvas id="series_of_bumps" width="340" height="300"></canvas> </center>
    </p>
    <p>Again, I've exaggerated the size of the windows of failure, in order
      to make them easier to see. It should be pretty clear that if we add
      all these bump functions up we'll end up with a reasonable
      approximation to $\sigma^{-1} \circ f(x)$, except within the windows
      of failure.</p>
    <p>Suppose that instead of using the approximation just described, we use
      a set of hidden neurons to compute an approximation to <em>half</em> our
      original goal function, i.e., to $\sigma^{-1} \circ f(x) / 2$. Of
      course, this looks just like a scaled down version of the last graph:</p>
    <p>
      <center> <canvas id="half_bumps" width="340" height="300"></canvas>
      </center>
    </p>
    <p>And suppose we use another set of hidden neurons to compute an
      approximation to $\sigma^{-1} \circ f(x)/ 2$, but with the bases of
      the bumps <em>shifted</em> by half the width of a bump:</p>
    <p>
      <center> <canvas id="shifted_bumps" width="340" height="300"></canvas>
      </center>
    </p>
    <p>Now we have two different approximations to $\sigma^{-1} \circ f(x) /
      2$. If we add up the two approximations we'll get an overall
      approximation to $\sigma^{-1} \circ f(x)$. That overall approximation
      will still have failures in small windows. But the problem will be
      much less than before. The reason is that points in a failure window
      for one approximation won't be in a failure window for the other. And
      so the approximation will be a factor roughly $2$ better in those
      windows.</p>
    <p>We could do even better by adding up a large number, $M$, of
      overlapping approximations to the function $\sigma^{-1} \circ f(x) /
      M$. Provided the windows of failure are narrow enough, a point will
      only ever be in one window of failure. And provided we're using a
      large enough number $M$ of overlapping approximations, the result will
      be an excellent overall approximation.</p>
    <p>
    <h3><a name="conclusion"></a><a href="#conclusion">Conclusion</a></h3>
    </p>
    <p>The explanation for universality we've discussed is certainly not a
      practical prescription for how to compute using neural networks! In
      this, it's much like proofs of universality for <tt>NAND</tt> gates and
      the like. For this reason, I've focused mostly on trying to make the
      construction clear and easy to follow, and not on optimizing the
      details of the construction. However, you may find it a fun and
      instructive exercise to see if you can improve the construction.</p>
    <p>Although the result isn't directly useful in constructing networks,
      it's important because it takes off the table the question of whether
      any particular function is computable using a neural network. The
      answer to that question is always "yes". So the right question to
      ask is not whether any particular function is computable, but rather
      what's a <em>good</em> way to compute the function.</p>
    <p>The universality construction we've developed uses just two hidden
      layers to compute an arbitrary function. Furthermore, as we've
      discussed, it's possible to get the same result with just a single
      hidden layer. Given this, you might wonder why we would ever be
      interested in deep networks, i.e., networks with many hidden layers.
      Can't we simply replace those networks with shallow, single hidden
      layer networks?</p>
    <p><span class="marginnote"><strong>Chapter
          acknowledgments:</strong> Thanks to <a href="http://jendodd.com">Jen Dodd</a> and <a
          href="http://colah.github.io/about.html">Chris Olah</a> for many
        discussions about universality in neural networks. My thanks, in
        particular, to Chris for suggesting the use of a lookup table to
        prove universality. The interactive visual form of the chapter is
        inspired by the work of people such as <a href="http://bost.ocks.org/mike/algorithms/">Mike Bostock</a>, <a
          href="http://www-cs-students.stanford.edu/&#126;amitp/">Amit
          Patel</a>, <a href="http://worrydream.com">Bret Victor</a>, and <a href="http://acko.net/">Steven Wittens</a>.
      </span></p>
    <p>While in principle that's possible, there are good practical reasons
      to use deep networks. As argued in
      <a href="chap1.html#toward_deep_learning">Chapter 1</a>, deep networks
      have a hierarchical structure which makes them particularly well
      adapted to learn the hierarchies of knowledge that seem to be useful
      in solving real-world problems. Put more concretely, when attacking
      problems such as image recognition, it helps to use a system that
      understands not just individual pixels, but also increasingly more
      complex concepts: from edges to simple geometric shapes, all the way
      up through complex, multi-object scenes. In later chapters, we'll see
      evidence suggesting that deep networks do a better job than shallow
      networks at learning such hierarchies of knowledge. To sum up:
      universality tells us that neural networks can compute any function;
      and empirical evidence suggests that deep networks are the networks
      best adapted to learn the functions useful in solving many real-world
      problems.
    </p>
    <p>
      <!-- Seems to be necessary to ensure the font loads --> <span style="font-family: MJX_Math; color: #fff;">.</span>
      <span style="font-family: MJX_Main; color: #fff;">.</span>
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      please cite this book as: Michael A. Nielsen, "Neural Networks and
      Deep Learning", Determination Press, 2015

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